In triangle $\Delta DEF$, we are given:

• $f = 33$ inches (side opposite $\angle F$),
• $\angle F = 140^\circ$,
• $\angle D = 5^\circ$.

We are tasked with finding the length of side $e$, which is opposite $\angle E$.

Step 1: Find $\angle E$

The sum of the angles in a triangle is always 180°. Using this information:

$\angle E = 180^\circ - \angle F - \angle D$ $\angle E = 180^\circ - 140^\circ - 5^\circ = 35^\circ$

So, $\angle E = 35^\circ$.

Step 2: Apply the Law of Sines

We can now use the Law of Sines to find the length of side $e$. The Law of Sines states:

$\frac{e}{\sin(\angle E)} = \frac{f}{\sin(\angle F)}$

Substitute the known values:

$\frac{e}{\sin(35^\circ)} = \frac{33}{\sin(140^\circ)}$

We know that $\sin(140^\circ) = \sin(40^\circ)$, since $\sin(180^\circ - x) = \sin(x)$.

So the equation becomes:

$\frac{e}{\sin(35^\circ)} = \frac{33}{\sin(40^\circ)}$

Step 3: Solve for $e$

Now, we can solve for $e$:

$e = \frac{33 \cdot \sin(35^\circ)}{\sin(40^\circ)}$

Using a calculator:

$\sin(35^\circ) \approx 0.5736 \quad \text{and} \quad \sin(40^\circ) \approx 0.6428$

Substitute these values:

$e = \frac{33 \cdot 0.5736}{0.6428} \approx \frac{18.943}{0.6428} \approx 29.5 \text{ inches}$

Final Answer:

The length of side $e$ is approximately 29.5 inches.

Would you like more details or have any questions?

Related Questions:

1. How do you calculate the third angle in a triangle?
2. How is the Law of Sines used in non-right triangles?
3. Can the Law of Cosines be used here, and if so, how?
4. What would happen if the angle sum condition is not satisfied in a triangle?
5. How do you calculate the area of a triangle when you know two sides and the included angle?

Tip: Always double-check your angle measurements when working with trigonometric functions in triangles.